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Theorem funimass3 5848
Description: A kind of contraposition law that infers an image subclass from a subclass of a preimage. Raph Levien remarks: "Likely this could be proved directly, and fvimacnv 5847 would be the special case of  A being a singleton, but it works this way round too." (Contributed by Raph Levien, 20-Nov-2006.)
Assertion
Ref Expression
funimass3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )

Proof of Theorem funimass3
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 funimass4 5779 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  ( F `  x )  e.  B ) )
2 ssel 3344 . . . . . 6  |-  ( A 
C_  dom  F  ->  ( x  e.  A  ->  x  e.  dom  F ) )
3 fvimacnv 5847 . . . . . . 7  |-  ( ( Fun  F  /\  x  e.  dom  F )  -> 
( ( F `  x )  e.  B  <->  x  e.  ( `' F " B ) ) )
43ex 425 . . . . . 6  |-  ( Fun 
F  ->  ( x  e.  dom  F  ->  (
( F `  x
)  e.  B  <->  x  e.  ( `' F " B ) ) ) )
52, 4syl9r 70 . . . . 5  |-  ( Fun 
F  ->  ( A  C_ 
dom  F  ->  ( x  e.  A  ->  (
( F `  x
)  e.  B  <->  x  e.  ( `' F " B ) ) ) ) )
65imp31 423 . . . 4  |-  ( ( ( Fun  F  /\  A  C_  dom  F )  /\  x  e.  A
)  ->  ( ( F `  x )  e.  B  <->  x  e.  ( `' F " B ) ) )
76ralbidva 2723 . . 3  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( A. x  e.  A  ( F `  x )  e.  B  <->  A. x  e.  A  x  e.  ( `' F " B ) ) )
81, 7bitrd 246 . 2  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A. x  e.  A  x  e.  ( `' F " B ) ) )
9 dfss3 3340 . 2  |-  ( A 
C_  ( `' F " B )  <->  A. x  e.  A  x  e.  ( `' F " B ) )
108, 9syl6bbr 256 1  |-  ( ( Fun  F  /\  A  C_ 
dom  F )  -> 
( ( F " A )  C_  B  <->  A 
C_  ( `' F " B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   A.wral 2707    C_ wss 3322   `'ccnv 4879   dom cdm 4880   "cima 4883   Fun wfun 5450   ` cfv 5456
This theorem is referenced by:  funimass5  5849  funconstss  5850  fvimacnvALT  5851  fimacnv  5864  r0weon  7896  iscnp3  17310  cnpnei  17330  cnclsi  17338  cncls  17340  cncnp  17346  1stccnp  17527  txcnpi  17642  xkoco2cn  17692  xkococnlem  17693  basqtop  17745  kqnrmlem1  17777  kqnrmlem2  17778  reghmph  17827  nrmhmph  17828  elfm3  17984  rnelfm  17987  symgtgp  18133  tgpconcompeqg  18143  eltsms  18164  ucnprima  18314  plyco0  20113  plyeq0  20132  xrlimcnp  20809  rinvf1o  24044  xppreima  24061  cvmliftmolem1  24970  cvmlift2lem9  25000  cvmlift3lem6  25013
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-fv 5464
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